Corrigendum: Strong $\mathbb A^1$-invariance of $\mathbb A^1$-connected components of reductive algebraic groups (J. Topol. 16 (2023), no. 2, 634--649.)

TL;DR

The corrigendum targets gaps in the original proof of strong $\mathbb A^1$-invariance for reductive algebraic groups and provides a complete, self-contained argument. It develops preliminaries on long exact sequences of homotopy groups for homotopy principal fibrations of simplicial sheaves, and constructs a key long exact sequence arising from a central isogeny $\widetilde{G} \to G$ with kernel $\mu$ of multiplicative type, relating Nisnevich and étale quotients via $Q$ and $\widetilde{Q}$ with $Q_{\text{et}} = B_{\text{et}}\mu$. The authors show that the relevant images in $\pi_0^{\mathbb A^1}$ are strongly $\mathbb A^1$-invariant, thereby bypassing the referenced gap and establishing the strong $\mathbb A^1$-invariance of $\pi_0^{\mathbb A^1}(G)$. This solidifies the $\mathbb A^1$-invariance framework for reductive groups and their torsors within Morel–Voevodsky theory, providing a reliable foundation for subsequent applications in motivic algebraic topology.

Abstract

The proof of Lemma 5.1 in the paper Strong $\mathbb A^1$-invariance of $\mathbb A^1$-connected components of reductive algebraic groups (J. Topol. 16 (2023), no. 2, 634--649) is incomplete as it relies on some results of Choudhury-Hagadi, the proof of which contains a gap. The goal of this note is to give a complete and self-contained proof of this lemma.

Theorems & Definitions (21)

• Theorem 1.1
• Proposition 2.1
• proof
• Corollary 2.2
• proof
• Proposition 2.3
• proof
• Remark 2.4
• Lemma 3.1
• proof